arising in periodic problems through the use of Ewald sums.

A closely related algebraic operation occurs in the evaluation of multilinear forms such as those that appear in density functional methods for calculating ground electronic states in quantum electronic structures. A bilinear form B((z_i), (x_i)) can always be written in terms of a matrix:

As written, the bilinear form would require O(n²) operations to evaluate. It can be computed instead by using the intermediate vector (y_i) defined above, calculation of which requires O(n) operations, as

This way of evaluating a bilinear form can greatly reduce overall computation time, depending on how efficiently y_i can be computed.

In density functional methods for calculating ground electronic states in quantum electronic structures, it is desired to approximate multilinear forms including integrals of the form

where

(Here, μ_i denotes a multi-index, say (p_i,q_i, r_i) so that

In particular, such a representation arises via an expansion

Hence,

The evaluation of ρ at a single point provides an example of the alternatives for evaluating a bilinear form. Using the representation involving the matrix (C_ijrequires n² operations, whereas the expression involving the χ's requires m·n operations. If m n, the latter approach will be more efficient.

The above integral can be viewed as a quadrilinear form involving the coefficients Multilinear forms can be evaluated in a variety of ways. It is tempting to represent them in terms of a precomputed tensor (a matrix for a bilinear form). Recently, Bagheri et al. (1994) have observed that it can be more efficient, in both time and memory, not to precompute expressions.

Front Matter (R1-R12)

Executive Summary (1-6)

Introduction (7-12)

The Emergence of Computational Chemistry (13-18)

Examples of Constructive Cross-Fertilization Between the Mathematical Sciences and Chemistry (19-42)

Mathematical Research Opportunities from Theoretical/Computational Chemistry (43-108)

Cultural Issues and Barriers to Interdisciplinary Work (109-116)

Conclusions and Recommendations (117-120)

Afterword (121-122)

Glossary (123-131)